DVI-I or DVI-D is appropriate. DVI-A is an analog connection and is similar to a VGA cable. DVI-D is a digital ONLY connection and is fine if you only plan on hooking up digital devices. DVI-I contains both digital and analog connections and allows you to do either analog or digital source components. Though, really, most will be digital, or DVI-D.
Yes, since you have a CRT television you need to be AWARE of possible burn in. Not concerned per-se, but aware. So, you want to limit your gaming, especially when you play a game that has a static image anywhere on the screen like the words "PLAYER 1" or a status map in the corner (Grand Theft Auto). Yep, just turn it off in an hour or so and you should be fine.
DLP and LCD do not have burn in issues, CRT and plasma do.